science

Unrestricted domain

Unrestricted domain is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Unrestricted domain rather than just read about it. In short: In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed. Intuitively, unrestricted domain is a common requirement for social choice functions, and is a condition for Arrow's impossibility theorem.

Key takeaways

Unrestricted domain belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Unrestricted domain to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Unrestricted domain from memory before moving on to harder problems.

Reference excerpt

In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed. Intuitively, unrestricted domain is a common requirement for social choice functions, and is a condition for Arrow's impossibility theorem. With unrestricted domain, the social welfare function accounts for all preferences among all voters to yield a unique and complete ranking of societal choices. Thus, the voting mechanism must account for all individual preferences, it must do so in a manner that results in a complete ranking of preferences for society, and it must deterministically provide the same ranking each time voters' preferences are presented the same way.

Relation to Arrow's impossibility theorem Unrestricted domain is one of the conditions for Arrow's impossibility theorem. Under that theorem, it is impossible to have a social choice function that satisfies unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non-dictatorship. However, the conditions of the theorem can be satisfied if unrestricted domain is removed.

Examples of restricted domains Duncan Black defined a restriction to domains of social choice functions called "single-peaked preferences". Under this principle, all of the choices have a predetermined position along a line, giving them a linear ordering. Every voter has some special place he likes best along that line. His ordering of the choices is determined by their distances from that spot. For example, if voting on where to set the volume for music, it would be reasonable to assume that each voter had their own ideal volume preference and that as the volume got progressively too loud or too quiet they would be increasingly dissatisfied. Black proved that by replacing unrestricted domain with single-peaked preferences in Arrow's theorem removes the impossibility: there are Pareto-efficient non-dictatorships that satisfy the "independence of irrelevant alternatives" criterion. However, Black's 1948 proof was published before Arrow's impossibility theorem was published in 1950, and thus did not fully consider Arrow's "unrestricted domain" criterion.

References Arrow, K.J. (August 1950), "A Difficulty in the Concept of Social Welfare" (PDF), Journal of Political Economy, 58 (4): 328–346, doi:10.1086/256963, S2CID 13923619.

Worked examples

Example 1 — a first encounter with Unrestricted domain

Start with the simplest possible case. Write down what Unrestricted domain claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Unrestricted domain before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Unrestricted domain ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Unrestricted domain

In research: Unrestricted domain appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Unrestricted domain in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Unrestricted domain is common in secondary-school and first-year university syllabi. It links to neighbouring topics Electoral system criteria, Political term stubs, Social choice theory, so understanding it makes those chapters shorter.
In everyday life: Look for Unrestricted domain outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Unrestricted domain in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Unrestricted domain means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Unrestricted domain out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Unrestricted domain in simple terms?

Why does Unrestricted domain matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Unrestricted domain?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Unrestricted domain.